The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$.
The proofs given in the book are fairly easy, using the axiom of choice. But then it is left as an exercise a generalization of the above theorems for not necessarily pairwise disjoint sets.
I have searched all over the web, but all the proofs found so far are for pairwise disjoint sets.
The proof given in the book relies on the fact that each set $A_k$ can be matched with a set like $[a_{k-1},a_k)$, with $a_k$ point of a partition of $[0,1]$, such that $\bigcup_{k=1}^{\infty\lor n}A_k=[0,1)$. So it is easy to guess the bijective function, but I can't generalize this proof.
Any help is highly appreciated. Thanks and regards.
Best Answer
HINT: Let $B_1=A_1$. For $n\ge 2$ let $B_n=A_n\setminus\bigcup_{k=1}^{n-1}A_k$. Then $\bigcup_nB_n=\bigcup_nA_n$, the sets $B_n$ are pairwise disjoint, each has cardinality at most $\mathfrak{c}$, and $B_1$ has cardinality $\mathfrak{c}$.